Inferential Statistics: Michaelis-Menten equation - Essay Example

 Inferential Statistics The Michaelis-Menten kinetics equation of enzymatic reactions V= Vmax [S Km + [S] (Sprott, 2001) Where; V representsthe speed/time of the enzymatic reaction while Vmax represents the velocity of the saturation of the enzyme with the substrate. [S] represents the substrate concentration and Km is referred to as the Michealis constant. He above equation has calculated error term denoted as e1 which is associated with deviations in the measured velocities (Sprott, 2001).

This paper assumes that this error is insignificant since it is statistically independent and it is a normal distribution within statistical mathematics. There has been much effort in determining the best estimations of Km and Vmax. Despite the fact that statistical limitations have been developed in terms Km and Vmax, there are still large inherent errors when it comes to estimation of these two parameters. Statistical calculation of the standard error when it comes to Km and Vmax has been neglected for long despite the availability of techniques of determining these parameters.

The importance of statistics of these two variables comes when identifying the competitive and non-competitive inhibition since without estimation of their errors it becomes impossible to identify confidently these inhibitors. There were difficulties in precise determination of Km and Vmax in the double reciprocal model of Michealis-Menten kinetics, and, therefore a larger inherent error in the two variables. This necessitated statistics researchers (Metzler, 1998) to come up with several improvements in order to reduce the error.

There was the Hanes-Woolf double reciprocal plot and then the Eadie-Hofstee semi-reciprocal plot. This paper will compare the estimation of Km and Vmax using simple Michealis-Menten simple kinetics double reciprocal equation and Eadie-Hofstee kinetics semi-reciprocal equation which is an improvement of the Michealis-Menten estimation of these parameters. The data below has been adopted from Atkinson, Jackson and Morton and was used before in statistical estimations of errors of Vmax and Km.

In the data set, the substrate concentration has been approximated in an inversely proportional way with almost similar data points above Km and below it. A table 1 showing Sample Enzymatic data which was used in analyzing the Michaelis-Menten Kinetics parameters Km and Vmax.Velocity VSubstrate Concentration [S]0.1480.1380.1710.2200.2340.2910.2340.560This is the appropriate position of Km0.3900.7660.4931.460Adopted from (Atkinson, Jackson and Morton, 1961) In an experiment in the literature by Raymond and Tania (2006), the analysis of the above data was carried out using statistical package FORTRAN for the double reciprocal Michael-Menten kinetics.

This R program employed calculated generalized least squares. This least-squire procedure searched for new estimates of KM and Vmax with the Sum of Square Residual (SSRes) reducing. Only the relative size of SSRes was used but not its magnitude. This R program calculated for given values of Vmax and Km the square of the differences of the observed V values and the calculated. This was then followed by the calculation of the SSRes.Formulae employed by the program;SSRes  (Atkinson, Jackson, Morton, 1961)The Lineweaver-Burk method in the determination of the Michaelis-Menten parameters; Km and VmaxThe Lineweaver method is a standard linear regression double reciprocal plot with the inverse of V (1/V) on the Y-axis and the inverse of [S] (1/[S]) on the X-axis.

Therefore;  = 0.798 + 1.708 (Raymond and Tania, 2006)Whereby; Regression (r) = 0.979 Probability (p) = 0.06%Therefore; 1/Vmax = 1.71+/- 0.303 (SE of mean) and Km/Vmax = 0.758+/- 0.0782 (SE of mean). The approximated inverse error (Z=1/X) can be calculated using the formulae below;SE (Z) = X-1 . On ignoring the covariance terms, the inverse error for a division (z = X/Y) can be obtained through the formula below;SE (Z) =  (Sprott, 2001)Using the above formulae of inversion error, the; Km SE (Standard Error) = 0.441 0.0906 and Vmax  SE= 0.585 0.104.

These estimates are imprecise despite the fact that the correlation coefficient (r) is 0.979 which shows a perfect fit. This is because on replacing the standard error by +/- 95% confidence interval through a multiplication by an appropriate t value (t = 2.78) (Metzler, 1998), the relative error of Km becomes +/- 57.3% while that of Vmax becomes +/- 49.2%. These are significantly big errors therefore, making the estimations of Km and Vmax of little or no utility. Analyzing the above enzymatic reaction data set with Eadie-Hofstee model This is a more straightforward method of determining the Michealis-Menten Km and Vmax .

This model does not require obtaining of the line of best fit through linear regression. The estimations of Km and Vmax using this model as obtained from the R program are; Km +/- SE = 0.491 +/- 0.083 and Vmax +/- 0.685 +/- 0.037. These estimates are for the un-weighted regression of V upon V/[S]. While the Km and Vmax estimations obtained from this method are close to those obtained from the double reciprocal model, the apparent errors of the two parameters are smaller.ReferencesAtkinson, M .

Jackson, J. and Morton, R. (1961) Statistical approach, Journal of Biochem 80, 318-323Metzler, C (1998) Statistical properties of kinetic estimates. In Kinetic Data Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments, London: Plenum PressRaymond, J. and Tania P. (2006) Current Statistical Methods for Estimating the Km and Vma~ of Michaelis-Menten Kinetics , Sydney: University of Sydney Press. Sprott, J. (2001) Numerical Recipes Routines and Examples in BASIC, Cambridge University Press, Cambridge